| L(s) = 1 | + (−2.51 − 4.35i)2-s + (−1.54 − 4.95i)3-s + (−8.65 + 14.9i)4-s + 0.150·5-s + (−17.7 + 19.2i)6-s + (−18.0 − 3.93i)7-s + 46.8·8-s + (−22.2 + 15.3i)9-s + (−0.378 − 0.654i)10-s + 46.9·11-s + (87.8 + 19.7i)12-s + (−8.75 − 15.1i)13-s + (28.3 + 88.7i)14-s + (−0.232 − 0.745i)15-s + (−48.7 − 84.3i)16-s + (−35.7 − 61.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.889 − 1.54i)2-s + (−0.298 − 0.954i)3-s + (−1.08 + 1.87i)4-s + 0.0134·5-s + (−1.20 + 1.30i)6-s + (−0.977 − 0.212i)7-s + 2.07·8-s + (−0.822 + 0.569i)9-s + (−0.0119 − 0.0207i)10-s + 1.28·11-s + (2.11 + 0.474i)12-s + (−0.186 − 0.323i)13-s + (0.542 + 1.69i)14-s + (−0.00400 − 0.0128i)15-s + (−0.761 − 1.31i)16-s + (−0.509 − 0.883i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.165830 + 0.129694i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.165830 + 0.129694i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.54 + 4.95i)T \) |
| 7 | \( 1 + (18.0 + 3.93i)T \) |
| good | 2 | \( 1 + (2.51 + 4.35i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 - 0.150T + 125T^{2} \) |
| 11 | \( 1 - 46.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + (8.75 + 15.1i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (35.7 + 61.9i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (57.4 - 99.4i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + 135.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (30.1 - 52.2i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (5.10 - 8.83i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-152. + 263. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (142. + 247. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (234. - 406. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (168. + 292. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-13.1 - 22.7i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-289. + 501. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (403. + 699. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-60.6 + 105. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 26.5T + 3.57e5T^{2} \) |
| 73 | \( 1 + (60.9 + 105. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-29.3 - 50.7i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-298. + 517. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-286. + 495. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (335. - 581. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.04259639071572368128093449197, −12.21215145691764586043334853059, −11.46519062156710541873385240547, −10.17740518446176119819672683069, −9.178196138260218349200144290679, −7.87106598057958693135355950530, −6.37002640578390145603720566668, −3.66622203898422609065725872490, −1.94416288825678814329247395126, −0.19512577064959666443390627549,
4.24403298932389688424928280083, 5.99433506329309056867374850439, 6.67053873493601231361749721555, 8.516944430749446799527412692526, 9.372205452924283588434818189053, 10.15474217852800493321436702821, 11.74493518469123604254809512700, 13.62753518412076966992713300259, 14.92387457237639201719240910973, 15.43182016840515760546439964242
